It is a bit childish. Being a native Dutch speaker I am quite puzzled about this prevalence of diminutives in duolingo. Of course there are quite some cases that Dutch would use the diminutive, for example 'telefoontje' to distinguish a phone call from a phone / 'telefoon', or 'kopje' for a cup (perhaps to distinguish it from 'kop' = head), in fixed expressions 'schouderklopje/ ik geef je op een briefje' and of course for people, animals or objects which are clearly much smaller than the norm. However, as a general rule it is in my opinion better to avoid diminutives unless you are quite sure that it is the right choice in the situation; overuse of diminutives, while used by children and some women, lowers your perceived status by quite a lot. Sentences like "Ik was in het winkeltje met die leuke gordijntjes en daar zag ik zo'n beeldig truitje dat ik gelijk mijn portemonneetje pakte en de eurootjes op het toonbankje neerlegde" is probably not something anyone would want to say outside a comedy sketch.

Of course, one should be able to recognize diminutives and be able to use them when needed, but something like 'vraagje', while theoretically possible in this context, is something I myself (male speaker) would never use, and I would not even recommend it to female speakers, possibly unless they are addressing another woman of higher status who dislikes them (or possibly children, who don't seem to mind too much). But by default, I would avoid diminutives unless there is a truly compelling reason to use them.

This is useful, thank you. In English I find myself saying things like "just a quick question" a lot to indicate that it is indeed going to be quick/simple/not important that the addressee has the answer for me. Would you use "vraagje" for that, or would you use some other construction to indicate the relative lack of gravity?

You should take several factors into account here. The most important factor in this sentence is that in Dutch, as in English, you can use extra politeness when directly addressing a person, but you generally don't use it when describing another person doing it. For example, if I would ask someone else of approximately my status to have a look at something, I'd say "Kan je hier even naar kijken" ('Could you take a brief look at it'), where 'even' (briefly/a bit) makes the formulation more polite, as does the "Kan" (can/could). But an observer would not say "Eric liet zijn teamleider even naar het document kijken" - he or she would say something like "De teamleider bestudeerde het document". (the team leader studied the document) or "Eric liet zijn teamleider het document bekijken" (Eric let his team leader take a look at the document." Note that in such formulations, politness words like 'kan' or 'even' disappear. While in some cases (where I will go into below) it would be okay (be it not preferable) to use 'vraagje' when asking the question yourself, using 'vraagje' in a third person context like above is, in my opinion, weird if not plain wrong.
And in directly addressing another, should one use 'vraagje'? Like I said, it depends. If you are a male, or a female who wants to project professionalism and/or competence, I would not recommend it. If the question is really short, you can use "snelle vraag"/"korte vraag". "Vraagje" itself can also be used if you are being sarcastic (if you are asking the question that will pop your enemy's bubble). In any case, wherever you can use 'vraagje', you can use 'vraag', and 'vraag' is usually if not always the better option. Hence my dislike for duolingo being so focused on exercising diminutives, that it seems to forget that the sentence should not only be grammatically correct, but also 'socially' correct Dutch or at least Dutch as it is used in normal situations. Learning a new language is hard enough - it's not really necessary to teach people bad linguistic habits.

I disagree with you that "vraagje" implies a lack of professionalism or bad linguistic habits. Sure in some situations you won't use it, but using it does not make you sound incompetent or unprofessional. If you come across as either of those it is caused by what your question is and how you formulate your question but it sure as hell is not caused by the use of "vraagje".

Apart from that I don't believe Duolingo is focused on exercising diminutives. However, since the Dutch language uses and abundance of diminutives it would be wrong to not expose people learning the language to them.

My apologies for being unclear - I meant that I have the impression that in some cases the Duolingo content developers are so focused on teaching grammatical constructs (in dedicated lessons) that they occasionally neglect the (admittedly somewhat vague) 'rules' of normal usage. For the rest (you seem to be a fellow Dutch native speaker) you are free to disagree with my assessment of the proper usage of diminutives. There are like 16 million of us nowadays. I would however politely suggest to take a look at a decent-quality Dutch newspaper (like Volkskrant or NRC) and count the fraction of nouns which are diminutives (outside the children's pages, of course). Of course, Dutch don't always talk like newspapers, but even in daily conversation and TV series and so on, the diminutive is in my experience not that much used outside truly unexpectely tiny things, certain fixed terms/expressions, some young animals, at kindergarten and primary schools, and at some clubs for rich wives. So yes, diminutives are used, but generally one needs a good reason to use them. And yes, just using 'vraagje' once is indeed not a big deal. But if you stuff your speech with diminutives, I do believe that you will be taken less seriously in professional contexts. I really don't hate diminutives, and it is definitely fine to expose students to them - I just want to make a plea for teaching/demonstrating appropriate or at least 'normal' usage of them. There are so many normal sentences that for one reason or another use diminutives, like "Het olifantje huppelde achter zijn moeder aan". "De meisjes holden naar school", "We maakten een ommetje". "De poes gaf een kopje tegen mijn been" and so on that it neither seems necessary nor advisable to use example sentences like "De student stelt een vraagje over de computer" which makes me squirm inside. Then again, I'm a 'native' to a certain Dutch subgroup (male, academically educated, in his thirties, Randstad) and cannot guarantee that 'vraagje' may actually be the better alternative for some contexts and some cultural subgroups (and for some people - whereas it is not very striking if a Dutch woman uses the word "beeldig", a Dutch male doing so would cause some eyebrows to be raised - and undoubtedly there are also words and phrases that sound weird when women use them) But based on my upbringing and experience, I can only advise this sentence to be replaced by a sentence that shows a more 'normal' diminutive use. After all, learning to recognize 'je/tje/pje' endings of words is really not extremely hard, so a handful of normal examples should be fine, and if the student encounters people using the word 'vraagje', they should have no trouble recognizing it even if they have trained on 'olifantje'. For the rest, thank you for your thoughts; it is always interesting to hear if/when a fellow native speaker disagrees with my experience, and to try find out why that is could be the case.

Sure there are sentences like this one where one in a normal situation would not use "vraagje", at least I wouldn't. But on any level, which has nothing to do with level of education, professionalism etc. it is perfectly normal to say for example ik heb een vraagje to indicate that they have a quick or perhaps not overly important question. It might depend on the kind of relationship you have with the person you are saying it to, but that does not mean that it won't be used by university professors or managers of big firms.

Apart from that, yes for us Dutch native speakers the use of diminutives and learning that -tje, -je, pje, kje indicates a diminutive it does not seem like a big deal. But seeing the questions people have for people trying to learn the language it Dutch diminutives can be confusing. Especially with words like ijsje, biertje, wijntje, telefoontje, kopje, etentje etc., which are diminutives but do not (have to) indicate that something is small/little and can even have quite a different meaning than the word they originate from.

I had a quick look and there are about 20 words in the Diminutive skill, many of which people already learned the non-diminutive. Of those there are, with a quick, count at least 14 that are used frequently from day to day without any childish connotation.